Question

If \({}^nC_x=56\) and \({}^nP_x=336\), then find n and x.

• A. 7, 3
• B. 8, 4
• C. 8, 3
• D. 9, 6

Hint:

nPx / nCx = x!; use that to find x first, then solve n(n-1)(n-2) = 336 for n.

Answer 1

The Correct Option is C

Step 1: Recall the relationship between permutations and combinations.
\({}^nP_x={}^nC_x\times x!\).

Step 2: Substitute the given values.
\(336=56\times x! \Rightarrow x!=\dfrac{336}{56}=6\).

Step 3: Solve for x.
Since \(3!=6\), \(x=3\).

Step 4: Solve for n using \({}^nC_3=56\).
\({}^nC_3=\dfrac{n(n-1)(n-2)}{6}=56 \Rightarrow n(n-1)(n-2)=336\). Testing \(n=8\): \(8\times7\times6=336\), which matches.

Step 5: Final Answer.
n = 8 and x = 3, so option C is correct.